Theorem of the highest weight

In representation theory, a branch of mathematics, the theorem of the highest weight states that the irreducible representations of semisimple Lie algebras or compact Lie groups are classified by their highest weights: given a simply-connected compact Lie group G with Lie algebra ${\mathfrak {g}}$, there is a bijection

$\lambda \mapsto [V^{\lambda }]$

from the set of integral points on the positive Weyl chamber, called dominant weights, to the set of equivalence classes of irreducible representations of the complexification of ${\mathfrak {g}}$ (or G); $V^{\lambda }$ is an irreducible representation with highest weight $\lambda$.

Statement

Let ${\mathfrak {g}}$ be a finite-dimensional semisimple complex Lie algebra with Cartan subalgebra ${\mathfrak {h}}$. Let

• $\Lambda$ = the ${\mathfrak {h}}$-weight lattice,
• $\Lambda _{\mathbb {R} }$ = the real vector space spanned by $\Lambda$,
• $\Phi ^{+}$ = set of positive roots, $\Phi ^{-}$ = the set of negative roots,
• ${\mathfrak {g}}={\mathfrak {g}}_{-}\oplus {\mathfrak {h}}\oplus {\mathfrak {g}}_{+}$ where ${\mathfrak {g}}_{-}$ is spanned by the ${\mathfrak {h}}$-weight vectors of negative weights and ${\mathfrak {g}}_{+}$ those of positive weights,
• $C=\{x\in \Lambda _{\mathbb {R} }\mid \langle x,\alpha \rangle \geq 0,\,\alpha \in \Phi ^{+}\}$, the positive Weyl chamber.

Then the theorem states:

• If V is a finite-dimensional irreducible representation of ${\mathfrak {g}}$, then the space of vectors v in V such that ${\mathfrak {g}}_{+}v=0$ has dimension 1; a non-zero vector that spans this one-dimensional space is called a highest weight vector of V and the weight of such a vector is called the highest weight of V.
• For a highest weight vector v of V, V is spanned by vectors obtained by applying elements of ${\mathfrak {g}}_{-}$ to v.
• Every highest weight is dominant in the sense that it lies in C.
• If two finite-dimensional irreducible representations have the same highest weight, they are equivalent.
• Given a dominant weight (i.e., an integral or lattice point of C), there exists a finite-dimensional irreducible representation whose highest weight is the given dominant weight.

The most difficult part is the last one; the construction of a finite-dimensional irreducible representation.

Proofs

There are at least three proofs:

• The theory of Verma modules contains the highest weight theorem. This is the approach taken in the standard textbooks (e.g., Humphreys and Dixmier).
• The Borel–Weil–Bott theorem constructs an irreducible representation as the space of global sections of an ample line bundle; the highest weight theorem results as a consequence. (The approach uses a fair bit of algebraic geometry but yields a very quick proof.)
• The invariant theoretic approach: one constructs irreducible representations as subrepresentations of a tensor power of the standard representions. This approach is essentially due to H. Weyl and works quite well for classical groups.

References

• Dixmier, Jacques (1996) [1974], Enveloping algebras, Graduate Studies in Mathematics, vol. 11, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0560-2, MR 0498740
• Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103.
• Humphreys, James E. (1972a), Introduction to Lie Algebras and Representation Theory, Birkhäuser, ISBN 978-0-387-90053-7.